/* qoj_p5247.cpp */
#include <bits/stdc++.h>

using i64 = long long;

int main() {
    std::cin.tie(nullptr)->sync_with_stdio(false);

    int n, k; std::cin >> n >> k;
    std::string s; std::cin >> s;

    std::vector<int> stk, b(n, n), id(n, -1); int cur = 0;
    for (int i = 0; i < n; ++i) {
        if (s[i] == '(') {
            stk.push_back(i);
        } else {
            if (!stk.empty()) b[i] = stk.back(), b[stk.back()] = i, stk.pop_back();
            else b[i] = -1;
        }
    }
    for (int i = 0; i < n; ++i) if (s[i] == '(' && b[i] < n && id[i] == -1) {
        for (int j = i; s[j] == '(' && j < n; j = b[j] + 1) id[j] = id[b[j]] = cur;
        ++cur;
    }

    // for (int i = 0; i < n; ++i) std::cerr << b[i] << ' ' << id[i] << '\n';

    auto calc = [&](int ql, int qr) {
        static int l = 0, r = 0; static i64 x = 0;
        static std::vector<int> f(cur);
        while (r < qr) x += s[r] == ')' && b[r] >= l ? ++f[id[r]] : 0, ++r;
        while (l > ql) --l, x += s[l] == '(' && b[l] < r ? ++f[id[l]] : 0;
        while (qr < r) --r, x -= s[r] == ')' && b[r] >= l ? f[id[r]]-- : 0;
        while (ql > l) x -= s[l] == '(' && b[l] < r ? f[id[l]]-- : 0, ++l;
        // assert(x == (qr - ql == 2 ? 1 : 0)); //

        // i64 y = 0;
        // for (int i = ql; i < qr; ++i) for (int j = i + 2; j <= qr; j += 2) {
        //     int x = 0; bool f = true;
        //     for (int k = i; k < j; ++k) x += (s[k] == '(' ? 1 : -1), f &= x >= 0;
        //     y += f && x == 0;
        // }
        // std::cerr << "Move " << l << ' ' << r << '\n';
        // if (y != x) {
        //     std::cerr << "X" << x << ' ' << y << '\n';
        //     for (auto &x : f) std::cerr << x << ' ';
        // }
        // assert(y == x);
        return x;
    };

    using Info = std::pair<i64, int>;
    auto merge = [&](const Info &x, const Info &y) {
        return Info{x.first + y.first, x.second + y.second};
    };
    auto get = [&](auto val) {
        std::vector<Info> f(n + 1, Info{1e18, -1}); f[0] = Info{0, 0};
        auto trans = [&](auto self, int l, int r, int ql, int qr) -> void {
            if (l >= r) return;
            int m = (l + r) / 2; std::pair<Info, int> x{Info{1e18, -1}, -1};
            assert(ql < qr);
            for (int i = ql; i < qr; ++i)
                x = std::min(x, {merge(f[i], Info{calc(i, m + 1) - val, 1}), i});
            f[m + 1] = std::min(f[m + 1], x.first);
            // std::cerr << "Got " << m + 1 << ' ' << x.second << ": "
            //     << x.first.first << ' ' << x.first.second << '\n';
            self(self, l, m, ql, x.second + 1), self(self, m + 1, r, x.second, qr);
        };
        auto solve = [&](auto self, int l, int r) -> void {
            if (l >= r) return;
            int m = (l + r) / 2;
            if (r - l > 1) self(self, l, m);
            // std::cerr << "Trans " << l << ' ' << m + 1
            //     << " -> " << m + 1 << ' ' << r + 1 << '\n';
            trans(trans, m, r, l, m + 1);
            if (r - l > 1) self(self, m, r);
        };
        solve(solve, 0, n);
        return f[n];
    };

    i64 lo = -1e10, hi = 0;
    while (lo < hi) {
        i64 m = (lo + hi + 1) / 2;
        auto [s, c] = get(m);
        // std::cerr << m << ": " << c << ' ' << s << '\n';
        if (c <= k) lo = m;
        else hi = m - 1;
        // std::cerr << lo << ' ' << hi << '\n';
    }

    auto [ans, c] = get(lo); ans += lo * k;
    std::cout << ans << '\n';

    return 0;
}